With more than 10,000 write-ins still to be counted, Murkowski is confident of victory because she's been getting nearly 90 percent of the write-in votes.
Murkowski would be the first candidate elected to the Senate elected as a write-in since Strom Thurmond in 1954.
CBS Evening News anchor Katie Couric spoke with Murkowski about her role in the federal government.
"This is a whole new world for me," says Murkowski. "And it'll be a whole new world for my colleagues in dealing with me."
Murkowski is back in Washington and this time she's making sure everyone knows how to spell her name.
Scroll down to see video of Katie Couric's interview with Sen. Murkowski.
"Running a write-in campaign against my party's wishes demonstrates that it's either political courage or just plain crazy," she says.
Beaten in the GOP primary by Tea Party favorite and Sarah Palin-endorsed Joe Miller, in some ways the loss was Murkowski's gain, breathing new life into a candidate who assured Alaskans that federal dollars would keep traveling north.
The Republican establishment supported Miller and now Murkowski is trying to figure out where she fits in.
Couric: "Will you feel a little like it's your first day of school and you don't know who your friends are going to be?"
Murkowski: "Maybe a little bit of that. Who are my friends? Who are my real friends?"
Part of a dying breed of moderate Republicans, Murkowski claims she's winning because she represents all Alaskans.
Murkowski: "I do not pass the purity test that the Tea Party has set out. It's just as simple as that. And that may be the case. But I don't think most people in my state pass that. There's a lot of people in Alaska that are pretty anti-government. But I think they would also agree that, 'Well, maybe the best thing is not that we shut government down.'"
And Murkowski says the message she's hearing loud and clear is compromise.
Murkowski: "I will tell you, I am not one of those who wants Obama to fail. If he does well, that means the country's doing well. We don't have time as a nation to spend all of what we do blocking. We have got to figure out how we get to a point where we can be sitting around the table and talking about these difficult problems and advancing some solutions."
She's not shy when it comes to speaking out about another Alaskan outdoorswoman but she minimizes the bad blood created when her father appointed her instead of Sarah Palin to serve out his term in the Senate.
Couric: "What's up with your relationship with Sarah Palin? Can you explain it?"
Murkowski: I'm still her senator. I'm going work hard to represent her, too. We don't really have much of a relationship."
Couric: "You have said you would not support Sarah Palin for president because she is not worldly enough."
Murkowski: "I just do not think that she has those leadership qualities, that intellectual curiosity that allows for building good and great policies. You know, she was my governor for two years. And I don't think that she enjoyed governing."
No matter what Palin does, Murkowski says 2012 is still a long way off and she's certain of one name that won't be on the ballot.
Couric: "Would you ever throw your hat in the ring? That would really shake things up, wouldn't it?"
Murkowski: "A national write-in campaign. Wouldn't that be wild? No, we're not…we're not thinking about this. At this point in time, I am still counting votes in Juneau."
And Murkowski met with Republican leader Mitch McConnell Monday to talk about her future. She thinks compromise with the majority Democrats is possible on energy policy. As ranking Republican on the energy committee she'll have a major role in trying to work out a deal.